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Theorem pnt 23555
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9595 . . . . . . 7  |-  1  e.  RR
21rexri 9646 . . . . . 6  |-  1  e.  RR*
3 1lt2 10702 . . . . . 6  |-  1  <  2
4 df-ioo 11533 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 11535 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 11360 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 11547 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,) +oo )  C_  ( 1 (,) +oo ) )
82, 3, 7mp2an 672 . . . . 5  |-  ( 2 [,) +oo )  C_  ( 1 (,) +oo )
9 resmpt 5323 . . . . 5  |-  ( ( 2 [,) +oo )  C_  ( 1 (,) +oo )  ->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
108, 9mp1i 12 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
118sseli 3500 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  ( 1 (,) +oo ) )
12 ioossre 11586 . . . . . . . . . . 11  |-  ( 1 (,) +oo )  C_  RR
1312sseli 3500 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
1411, 13syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
15 2re 10605 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 11321 . . . . . . . . . . 11  |- +oo  e.  RR*
17 elico2 11588 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) ) )
1815, 16, 17mp2an 672 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) )
1918simp2bi 1012 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
20 chtrpcl 23205 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 661 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
22 0red 9597 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  e.  RR )
231a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  e.  RR )
24 0lt1 10075 . . . . . . . . . . . 12  |-  0  <  1
2524a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  <  1 )
26 eliooord 11584 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
2726simpld 459 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  < 
x )
2822, 23, 13, 25, 27lttrd 9742 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  0  < 
x )
2913, 28elrpd 11254 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
3011, 29syl 16 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
3121, 30rpdivcld 11273 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  /  x )  e.  RR+ )
3231adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
33 ppinncl 23204 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3414, 19, 33syl2anc 661 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
3534nnrpd 11255 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
3613, 27rplogcld 22770 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  ( log `  x )  e.  RR+ )
3711, 36syl 16 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
3835, 37rpmulcld 11272 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
3921, 38rpdivcld 11273 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
4039adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4130ssriv 3508 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR+
42 resmpt 5323 . . . . . . . 8  |-  ( ( 2 [,) +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) ) )
4341, 42ax-mp 5 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) )
44 pnt2 23554 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
45 rlimres 13344 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
4644, 45mp1i 12 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,) +oo )
)  ~~> r  1 )
4743, 46syl5eqbrr 4481 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
48 chtppilim 23416 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
4948a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
50 ax-1ne0 9561 . . . . . . 7  |-  1  =/=  0
5150a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
5239rpne0d 11261 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
5352adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5432, 40, 47, 49, 51, 53rlimdiv 13431 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5514recnd 9622 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
56 chtcl 23139 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5713, 56syl 16 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  RR )
5857recnd 9622 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  CC )
5911, 58syl 16 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
6055, 59mulcomd 9617 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  x.  ( theta `  x
) )  =  ( ( theta `  x )  x.  x ) )
6160oveq2d 6300 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6238rpcnd 11258 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
6330rpne0d 11261 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  =/=  0 )
6421rpne0d 11261 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
6562, 55, 59, 63, 64divcan5d 10346 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6661, 65eqtrd 2508 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6738rpne0d 11261 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
6859, 55, 59, 62, 63, 67, 64divdivdivd 10367 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
6934nncnd 10552 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
7037rpcnd 11258 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
7137rpne0d 11261 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
7269, 55, 70, 63, 71divdiv2d 10352 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  x ) )
7366, 68, 723eqtr4d 2518 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
7473mpteq2ia 4529 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
75 1div1e1 10237 . . . . 5  |-  ( 1  /  1 )  =  1
7654, 74, 753brtr3g 4478 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7710, 76eqbrtrd 4467 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
78 ppicl 23161 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
7913, 78syl 16 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  NN0 )
8079nn0red 10853 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  RR )
8129, 36rpdivcld 11273 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
8280, 81rerpdivcld 11283 . . . . . . 7  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  RR )
8382recnd 9622 . . . . . 6  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  CC )
8483adantl 466 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
85 eqid 2467 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8684, 85fmptd 6045 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,) +oo ) --> CC )
8712a1i 11 . . . 4  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR )
8815a1i 11 . . . 4  |-  ( T. 
->  2  e.  RR )
8986, 87, 88rlimresb 13351 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  |`  (
2 [,) +oo )
)  ~~> r  1 ) )
9077, 89mpbird 232 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9190trud 1388 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 973    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505    |` cres 5001   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    x. cmul 9497   +oocpnf 9625   RR*cxr 9627    < clt 9628    <_ cle 9629    / cdiv 10206   NNcn 10536   2c2 10585   NN0cn0 10795   RR+crp 11220   (,)cioo 11529   [,)cico 11531    ~~> r crli 13271   logclog 22698   thetaccht 23120  πcppi 23123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-o1 13276  df-lo1 13277  df-sum 13472  df-ef 13665  df-e 13666  df-sin 13667  df-cos 13668  df-pi 13670  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-cxp 22701  df-em 23078  df-cht 23126  df-vma 23127  df-chp 23128  df-ppi 23129  df-mu 23130
This theorem is referenced by: (None)
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